A simple function is a finite sum 
, where the functions 
are characteristic
functions of subsets 
of a set 
. Another description of a simple function is a function
that takes on finitely many values in its range.
The collection of simple functions is closed under addition and multiplication.
To define the Lebesgue integral, the subsets 
must be measurable
sets. A nonnegative measurable
simple function on a measure space has integral 
when the 
are pairwise disjoint sets.
Approximating a nonnegative measurable
function 
by nonnegative measurable
simple functions defines its Lebesgue integral.
